Exercise 58, § 1

a) Let $\varepsilon(T)$ be the formal series $\exp(- \sum_{n=0}^\infty T^{p^n}/p^n)$ of $\mathbf{Q}[[T]]$. We have

where $\mu$ is the Möbius function. The coefficients of $\varepsilon(T)$ belong to $\mathbf{Z}_{(p)}$. b) Let $X = (X_n)_{n \in \mathbf{N}}$ be a family of indeterminates. In the algebra $\mathbf{Q}[(X_n)_{n \in \mathbf{N}}]$, we have the equality c) Let $A$ be a $\mathbf{Z}_{(p)}$-algebra. Suppose that multiplication by $p.1_A$ is injective in $A$. Let $a = (a_n)_{n \in \mathbf{N}} \in A^\mathbf{N}$. For the series $\exp(\sum_{n=0}^\infty a_n T^{p^n}/p^n)$ of $A\left[\frac{1}{p}\right][[T]]$ to have its coefficients in $A$, it is necessary and sufficient that $a$ belong to $\Phi_A(W(A))$. d) Let $A$ be a $\mathbf{Z}_{(p)}$-algebra. The mapping $E \circ \omega_A$ of $W(A)$ into $\Lambda(A)$ (cf. p. 54, Exercise 40 and p. 57, Exercise 43) associates with $a = (a_n)_{n \in \mathbf{N}}$ the element $\prod_{n=0}^\infty \varepsilon(a_n(-T)^{p^n})$ of $\Lambda(A)$. e) Let $A$ be a $\mathbf{Z}_{(p)}$-algebra. Suppose that multiplication by $p.1_A$ is injective in $A$. Let $\sigma$ be an endomorphism of $A$ satisfying $\sigma a \equiv a^p \mod pA$. Let $s_\sigma : A \to W(A)$ be the homomorphism associated with $\sigma$ (p. 44, Exercise 14). Then, for every $a \in A$, the formal series has its coefficients in $A$, and we have f) Let $\mathscr{A}$ be the ring $\mathbf{Z}_{(p)}[(X_n)_{n \in \mathbf{N}}]$ and let $X = (X_n)_{n \in \mathbf{N}} \in W(\mathscr{A})$. The series $E_{F_{\mathscr{A}}}(X, T)$ has its coefficients in $W(\mathscr{A})$. For every $\mathbf{Z}_{(p)}$-algebra $A$ and every element $a = (a_n)_{n \in \mathbf{N}}$ of $W(A)$, we denote by $E_F(a, T)$ the series obtained by applying $W(\varphi)$ to the coefficients of $E_{F_{\mathscr{A}}}(X, T)$, where $\varphi : \mathscr{A} \to A$ is the mapping which associates $a_n$ with $X_n$. Then $E_F(a, T)$ has its coefficients in $W(A)$ and if $s_A$ denotes the homomorphism $W(A) \to W(W(A))$ defined in Exercise 15, p. 44, we have